Have you tried to connect the external hard drive on different computers with different operating systems recently? If so, tell me more about it. I am also interested if the external drive has additional power cord or it receives the power by an USB cable. I'd appreciate if you give me more details about that.
If you have the option, I recommend you go through the Control Panel> Computer Management> Disk Management and tell me if the external drive is recognized and has a drive letter assigned to it. If not, you have to assign a drive letter manually. Make sure you don't format the drive as you would lose all of the data.
If the hard drive is not seen in My Computer, you can reboot your machine while the disk is plugged in and enter the BIOS menu to see if the system can recognize at a physical level. Can you hear the hard drive spinning while connected or everything is calm and does not feel any vibration when you touch the case? If so, the disk must be brought to a data recovery company in order to help you extracting the information.